Method for the identification of weak and/or strong nodes of an electric power system

ABSTRACT

The subject of the invention is a method for the identification of weak and/or strong nodes of an electric power transmission system.  
     The inventive method consists in subjecting the electric parameters characterising nodes and branches of an electric power transmission system to computational treatment in order to obtain equations of power flow in all nodes of the system at assumed 100 percent system load value. Then earth is taken as the reference point, and nodal impedance values are computed for all nodes of the system, which values are then used to construct a P-Q curve which presents the functional relation between the system&#39;s reactive and active load. In the next operation, the nodal coefficient of voltage stability is determined as the relative distance between the base load point of that node and the critical point on the P-Q curve which is situated most closely to the base point. The numerical value of this coefficient is compared with a threshold value considered to be a safe margin to maintain voltage stability for a given node.

[0001] The subject of the invention is a method for the identification of weak and/or strong nodes of an electric power transmission system comprising at least one generator and nodes, interconnected by transmission lines, useful especially for the determination of the weak nodes of the analysed system. The method for the determination of the weak nodes of a power system employs known methods of determining the voltage stability of the whole system and predicts the voltage stability margin in specific nodes of the power system.

[0002] From U.S. Pat. No. 5,745,368 there is known a method of voltage stability analysis in electric power systems. That description discloses a method which is appropriate for low and high voltage applications as well as differing types of loads and load variations. In that method, a nose point of a P-Q curve showing functional relation between voltage and power is found, from which the distances to points characterizing the reactive, active and apparent power are calculated, while a generalised curve fit is used to compute the equivalent or surrogate nose point. The determination of that point is achieved by approximating a stable branch and creating a voltage versus power curve, determining a plurality of stable eqilibrium points on the voltage and load curve, using the plurality of determined stable eqilibrium points to create and fit an approximate stable branch, calculating an approximate voltage collapse point and thereafter a voltage collapse index. The value of that index allows for predicting the occurrence of expected voltage collapse under specific conditions.

[0003] From a European patent application No. EP 1 134 867 there is known a method for the assessment of stability of electric power transmission networks. The method comprises the measurement of vector quantities for voltage and current in numerous points in the network, transfer of those data to the system protection centre, transfer of information regarding the operating condition of equipment in the substations of that network, and, on the basis of the acquired data, determination of at least one stability margin value of the transmission network. The measured vectors may be represented by quantities such as voltage, current, power, or energy connected with phase conductor or an electronic system.

[0004] The method for the identification of weak and/or strong nodes of an electric power system according to the invention can be possibly employed as a useful solution for the assessment of stability of power networks, for example, in the solution presented in the application EP 1 134 867, although the identification of weak nodes in networks is made apart from the methods of network stability assessment as presented in the state of the art, and the method as such is not yet known.

[0005] On the other hand, from U.S. Pat. No. 5,796,628 there is known a dynamic method for preventing voltage collapse in electric power systems. In the presented solution “weak areas” in networks are identified. These areas are defined as those parts of the network which do not withstand additional load. The solution introduced in that description consists in monitoring the power network through the surveillance of real-time data from the network, forecasting the near-term load of each branch of the network as well as the power demand in that branch on the basis of those data, and in order to estimate the system stability, such that each of the branches would be able to withstand the expected load, the amount of the margin of reactive and/or active load is defined. The proposed value of this load as well as the proposed voltage profiles are determined on the basis of the known power flow technique and the saddle-node bifurcation theory.

[0006] The method for the identification of weak and/or strong nodes of an electric power transmission system according to the invention, which employs known computational methods regarding power flow in the nodes and branches of an electric power transmission system, and in which functional relations between active and reactive loads for that system are analysed, consists in subjecting the characteristic electric parameters of nodes and branches of the power system to computational treatment to achieve power flow equations for all that system's nodes with assumed 100 percent system load value, which treatment is used to calculate the complex voltage values in those nodes. Then the ground is assumed as the reference node and the values of node impedance for all nodes of the system are calculated, which values are then used to construct the P-Q curve which shows functional relation between reactive and active load of the system. Individually for each node, in the next step, the nodal voltage stability coefficient is determined as the relative distance between the base load point of this node and the critical point on the P-Q curve most closely situated to that base point, and then the numerical value of this coefficient is compared to the threshold value considered to be a safe margin for maintaining voltage stability for the given node. The result of this comparison allows to identify the analysed node as weak or strong.

[0007] Preferably the nodal voltage stability coefficient is calculated from this relation: $k_{cr} = {\frac{\sqrt{\left( {p_{cr} - p_{b}} \right)^{2} + \left( {q_{cr} - q_{b}} \right)^{2}}}{\sqrt{\left( p_{cr} \right)^{2} + \left( q_{cr} \right)^{2}}}\quad {,\quad}}$

[0008] where:

[0009] p_(cr)—is the values of the coordinates of active load in the node during critical operating conditions at the voltage stability limit,

[0010] q_(cr)—is the values of the coordinates of reactive load in the node during critical operating conditions at the voltage stability limit,

[0011] p_(b)—is the values of the coordinates of the base point of active load in the analysed node,

[0012] a q_(b)—is the values of the coordinates of the base point of reactive load in the analysed node.

[0013] Preferably the analysed node is considered to be weak where the value of the nodal voltage stability coefficient is less than or equal to 0.8 or strong where the value of the nodal voltage stability coefficient is bigger than 0.8.

[0014] The advantage of the method according to the invention is the ability to determine the weak and/or strong nodes of an electric power transmission system without the need for making a multivariant analysis of power flow in the power system considering the critical loads and cutouts of individual system elements.

[0015] The method according to the invention will be presented more closely by its embodiment and a drawing, where

[0016]FIG. 1 shows the schematic diagram of the power system structure,

[0017]FIG. 2—schematic diagram of the power system structure with altered reference node,

[0018]FIG. 3—exemplary diagram showing the relation between active load P and reactive load Q for a receiver node with marked base load point and a critical point, and

[0019]FIG. 4—a set of operations necessary to realise this method.

[0020] In a schematic presentation in FIG. 1 the electric power transmission system is a network formed by feed generators G connected with generator nodes W_(G) which in turn are connected to at least one receiver node W_(O) by means of appropriate transmission lines. At least one of the generator nodes W_(G) is connected through a transmission line with a flow node W_(S), which in turn is connected to at least one receiver node W_(O). Further on in the description, all transmission lines are called system branches.

[0021] For the network system formed as shown above, in the first stage of the realisation of the method, electric parameters in the system's nodes and in its branches are measured. In generator G voltage E_(G) is measured. In generator nodes W_(G) voltage V_(G) and active load P_(G) are measured In receiver nodes W_(O) voltage V_(O), active load PO and reactive load Q_(O) are measured. In flow node W_(S) voltage V_(S) is measured. In branches connecting the analysed generator nodes W_(G) with the flow node W_(S) and with receiver nodes W_(O) resistance R_(b), reactance X_(b) and susceptance B_(b) are measured.

[0022] Measurement data are fed to the control device, not shown in the drawing, which is a computer provided with suitable software, where the data are stored in its memory in a suitable digital form.

[0023] Measurement data recorded in the control device are made complete when the device reads-in the sychronous reactance x_(G) of generator G and its apparent power S_(nG). Operations regarding data preparation are shown in FIG. 4 as block 1.

[0024] When all the necessary data have been collected, the control device computes the equations of power flow in all nodes W_(G), W_(O), and in node W_(S) of the system, using known mathematical methods suitable for such purposes, such as, for instance, the Newton's method. For the computation, 100% total system load is assumed. The result of the conducted calculations concerning power flow are complex values of voltages in all nodes of the system.

[0025] The computing operations concerning the standard calculation of power flow, with a 100% system load, are shown in FIG. 4 as block 2.

[0026] Then, in stage two, /FIG. 2/ the earth is assumed as a reference node W_(Z) with respect to which further actions are carried out. These actions consist in modelling the node load in the form of its node admittance and determining the Kirchoff's and Ohm's equations matrix comprising the determined synchronous reactances x_(G) of generator G and the complex voltage values determined earlier for 100% load. $\begin{bmatrix} {E_{G}\frac{1}{x_{G}}} \\ 0 \end{bmatrix} = {{Y_{n}\begin{bmatrix} V_{G} \\ V_{O} \end{bmatrix}}{\quad,}}$

[0027] where Y_(n) is the node admittance matrix.

[0028] In the next step, node impedance matrix Z_(n) is determined as the inverse of matrix Y_(n) and after transformation the following matrix equation is obtained: $\begin{bmatrix} V_{G} \\ V_{O} \end{bmatrix} = {{Z_{n}\begin{bmatrix} {E_{G}\frac{1}{x_{G}}} \\ 0 \end{bmatrix}}\quad.}$

[0029] From the above matrix equation the equation describing operating conditions for the k^(th) receiver node is determined as:

V _(Ok) =Z _(k1) E _(G1) +. . . +Z _(ki) E _(Gi) +. . . +Z _(kk) O _(Ok) +. . . +Z _(kn) O _(On)  /1/.

[0030] Then a small variation in active load ΔP and reactive load ΔQ in the examined receiver node W_(O) is assumed, which variation produces small current increases ΔI_(O) and voltage increases ΔV_(O) in this node. This variation is modelled by connecting additional impedance Z_(aOk) in the k^(th) receiver node W_(O). For changed load conditions the equation /1/ will have the following form:

V _(Ok) +ΔV _(Ok) =Z _(k1) E _(G1) +. . . +Z _(ki) E _(Gi) +. . . +Z _(kk) ΔI _(Ok) +. . . +Z _(kn) O _(On)  /2/.

[0031] Hence, having transformed the equations /1/and /2/we receive an equation defining the value of node impedance in the k^(th) receiver node, which has this form:

Z _(kk) =ΔV _(OK) /ΔI _(OK)  /3/,

[0032] where

[0033] ΔV_(OK)—is voltage increment in the k^(th) node accompanying a load change,

[0034] ΔI_(OK)—is current increment in the k^(th) node accompanying a load change. It follows from the equation /3/that having measured the values of voltages and currents in the k^(th) receiver node before and after voltage variation one can determine the values of node impedance Z_(kk).

[0035] In the next step of the procedure, all operations performed following the change of the reference node shall be repeated, for all generator nodes W_(G) of the system, the estimated node impedance being the synchronous impedance Z_(Gi) of generator G, which is:

Z _(Gi) =ΔV _(Gi) /ΔI _(Gi)  /4/,

[0036] where

[0037] ΔV_(Gi)—is the voltage increment in the i^(th) generator node accompanying load change,

[0038] ΔI_(Gi)—is the current increment in the i^(th) generator node accompanying load change.

[0039] Knowing the values of node impedances Z_(kk), generator synchronous impedances Z_(Gi), voltage in receiver nodes V_(Ok), current in receiver nodes I_(Ok) and generator voltage E_(Ok), the power flow equation can be presented in the following complex form: $\begin{matrix} {{\underset{\_}{S}}_{Ok} = {{\left( \frac{1}{{\underset{\_}{Z}}_{kk}} \right)^{*}V_{Ok}^{2}} + {\left( \frac{- 1}{{\underset{\_}{Z}}_{kk}} \right)^{*}{\underset{\_}{E}}_{Tk}^{*}\underset{\_}{V}{\quad,}}}} & |5| \end{matrix}$

[0040] where:

E _(Tk) =V _(Ok) +V _(Ok) Z _(kk)  /6/,

[0041] which, following a number of transformations, can be written in the form of this equation:

(−xp+rq)² −rp−xq−0.25=0  /7/,

[0042] where: “p” and “q” are the variables of this equation, and “x” and “r” are coefficients defining the shape of the curve P-Q.

[0043] Equation /7/presents the curve P-Q as a functional relation between active load P and reactive load Q in a given, analysed receiver node, in a cartesian coordinate system, as in FIG. 3.

[0044] The above mentioned operations, consisting in assuming the earth as the reference node, and computing node impedances both for receiver nodes and generator nodes, and plotting the curve P-Q are marked in FIG. 4 as block 3.

[0045] Next, for base load point N, indicated in the coordinate system with the curve P-Q, defined by coordinates (p_(b),q_(b)), which characterizes the base load of the node, we determine the minimum distance between this point and critical point C of coordinates (p_(cr), q_(cr)) situated on the previously plotted curve P-Q.

[0046] The operation of determining the load point is marked in FIG. 5 as block 4.

[0047] In the next step, on the basis of the reciprocal positions of point N, which reflects the base load defined by coordinates (p_(b),q_(b)), point C of coordinates (p_(cr), q_(cr)) situated on curve P-Q, positioned at a minimum distance from point N, and the point defining the beginning of the coordinate system in which the curve P-Q has been plotted, the nodal coefficient of voltage stability k_(cr) is determined as the relative distance between points N and C and its value is calculated from this equation: $\begin{matrix} {k_{cr} = {\frac{\sqrt{\left( {p_{cr} - p_{b}} \right)^{2} + \left( {q_{cr} - q_{b}} \right)^{2}}}{\sqrt{\left( p_{cr} \right)^{2} + \left( q_{cr} \right)^{2}}}\quad {,\quad}}} & |8| \end{matrix}$

[0048] where:

[0049] p_(cr)—is the values of coordinates of active load in the node during critical operating conditions at the voltage stability limit,

[0050] q_(cr)—is the values of coordinates of reactive load in the node during critical operating conditions at the voltage stability limit,

[0051] p_(b)—is the values of coordinates of active load base point in the analysed node,

[0052] a q_(b)—are the values of coordinates of reactive load base point in the analysed node.

[0053] Calculation of the nodal coefficient of voltage stability is indicated as block 5 in FIG. 4.

[0054] In the next step, presented in FIG. 4 as block 6, identification of the system's nodes is made by comparing the numerical value of the k_(cr) coefficient determined for the given node with the assumed threshold value equal to 0.8, which value assumes a 20% allowance as a safety margin to maintain the node voltage stability. The result of the comparison serves to determine whether the analysed node is weak, for instance, when the value k_(cr)≦0.8, or if the node is strong, when k_(cr)>0.8. 

1. A method for the identification of weak and/or strong nodes of an electric power transmission system, which employs known analytical methods appropriate to power flow in nodes and branches of the power system and which analyses functional relations between active and reactive load for this system, characterised in that the electric parameters characterising the nodes and branches of the power system are subjected to computational treatment in order to obtain power flow equations in all nodes of the system at assumed 100 percent system load, which treatment is used to calculate complex voltage values in those nodes, and thereafter the earth is taken as a reference node, and the node impedance values for all the system's nodes are calculated, which values are then used to construct a curve /P-Q/ showing the functional relation between reactive load /Q/ and active load /P/ of the system, and then for each node individually the nodal coefficient of voltage stability is determined as the relative distance between the base load point of that node and the critical point on the curve /P-Q/ which is situated most closely to the base point, whereupon the numerical value of that coefficient is compared with a threshold value considered to be a safe margin to maintain voltage stability for the given node, and on the basis of that comparison, the analysed node is identified as weak or strong.
 2. The method according to claim 1, characterised in that the nodal coefficient of voltage stability /k_(cr)/ is calculated from this relation: $k_{cr} = {\frac{\sqrt{\left( {p_{cr} - p_{b}} \right)^{2} + \left( {q_{cr} - q_{b}} \right)^{2}}}{\sqrt{\left( p_{cr} \right)^{2} + \left( q_{cr} \right)^{2}}}\quad {,\quad}}$

where: p_(cr)—is the values of active load coordinates in the node during critical operating conditions at voltage stability limits, q_(cr)—is the values of reactive load coordinates in the node during critical operating conditions at voltage stability limits, p_(b)—is the values of the coordinates of the base point of active load in the analised node, and q_(b)—is the values of the coordinates of the base point of reactive load in the analised node.
 3. The method according to claim 1 or 2, characterised in that the analysed node is considered weak where the value of the nodal stability coefficient /k_(cr)/ is less than or equal to 0.8 or it is considered strong when the value of the nodal stability coefficient /k_(cr)/ is more than 0.8. 